Question

1. for each question, use the diagram to fill in the blanks to complete the proof.

a. given: ( m parallel n )
prove: ( angle 1 cong angle 5 )
b. given: ( m parallel n )
prove: ( angle 3 cong angle 6 )
c. given: ( m parallel n )
prove: ( angle 2 cong angle 7 )
d. given: ( angle 2 cong angle 7 )
prove: ( m parallel n )

Explanation:

Step1: Use the property of parallel lines and corresponding angles

When \(m\parallel n\), \(\angle1\) and \(\angle5\) are corresponding angles. By the Corresponding Angles Postulate, if two parallel lines are cut by a transversal, then corresponding angles are congruent. So for part a:
Statements: \(\angle1\cong\angle5\)
Reasons: Corresponding Angles Postulate

Step2: Use the property of parallel lines and alternate - interior angles

When \(m\parallel n\), \(\angle3\) and \(\angle6\) are alternate - interior angles. By the Alternate - Interior Angles Theorem, if two parallel lines are cut by a transversal, then alternate - interior angles are congruent. So for part b:
Statements: \(\angle3\cong\angle6\)
Reasons: Alternate - Interior Angles Theorem

Step3: Use the property of parallel lines and alternate - exterior angles

When \(m\parallel v\), \(\angle2\) and \(\angle7\) are alternate - exterior angles. By the Alternate - Exterior Angles Theorem, if two parallel lines are cut by a transversal, then alternate - exterior angles are congruent. So for part c:
Statements: \(\angle2\cong\angle7\)
Reasons: Alternate - Exterior Angles Theorem

Step4: Use the property of congruent angles and parallel lines (converse)

Given \(\angle2\cong\angle7\). \(\angle2\) and \(\angle7\) are alternate - exterior angles. By the Converse of the Alternate - Exterior Angles Theorem, if two lines are cut by a transversal and alternate - exterior angles are congruent, then the lines are parallel. So for part d:
Statements: \(m\parallel n\)
Reasons: Converse of the Alternate - Exterior Angles Theorem

Answer:

a. Statements: \(\angle1\cong\angle5\); Reasons: Corresponding Angles Postulate
b. Statements: \(\angle3\cong\angle6\); Reasons: Alternate - Interior Angles Theorem
c. Statements: \(\angle2\cong\angle7\); Reasons: Alternate - Exterior Angles Theorem
d. Statements: \(m\parallel n\); Reasons: Converse of the Alternate - Exterior Angles Theorem